Part 3: Calculating formation energies

The formation energy of a molecular species/configuration is the energy required for (or released while) generating that configuration from molecules within the reference set. We must note that what we refer to as “formation energy”, is different than the (Gibbs) free energy of formation, or the heat (enthalpy) of formation, used in Physical Chemistry. The free energy and the enthalpy include contributions from translational, rotational, and vibrational degrees of freedom. In our case, we work exclusively with energies provided by a DFT code, which are potential energies within the framework of the Born-Oppenheimer approximation. Roughly speaking, the latter allows us to fix the positions of the nuclei and calculate the energy of the system as the sum of the kinetic energy of the electronic cloud and the energies from the Coulomb interactions between electrons and nuclei.

Going back to the calculation of the formation energies, by the definition of the previous paragraph, it is clear that the formation energy of any reference species is zero. On the other hand, the energy of an adsorbed species, for instance CO* would be:

FE[CO*] = E[CO*] – { E[Pt(111)] + E[CO] }

In the above equation, FE[A] stands for formation energy of species A, E[A] means the DFT-computed energy of A. Thus to compute the formation energy of CO*, we take the DFT energy of CO* and subtract the DFT energies of pristine Pt(111) and CO gas. This was an easy example, since CO gas is a reference species. What about the energy of COOH*, as species which is not within the reference set? Its formation energy is defined as:

FE[COOH*] = E[COOH*] – { E[Pt(111)] + E[CO] + E[H2O] – ½ E[H2] }

Notice that we calculated the COOH* species formation energy with respect to a linear combination of reference species energies. This linear combination is determined by the composition stoichiometry: one CO plus one H2O minus half an H2 gives us precisely the same number of C, O, H atoms as in the COOH molecule. For a proper reference set, this linear combination is unique, making the definition of formation energies unambiguous.

In the table below we present the formation energies of all species calculated in a similar way as outlined above. Highlighted in light red are the DFT energies of reference species.

Catalytic Surface DFT Energy (eV) Form. Energy (eV)
Pt(111) -11448.220 0.000
Gas Species DFT Energy (eV) Form. Energy (eV)
CO -588.789 0.000
H2O -467.460 0.000
H2 -31.403 0.000
CO2 -1025.462 -0.615
O2 -867.202 4.913
Surface Species DFT Energy (eV) Form. Energy (eV)
CO* -12039.087 -2.077
H2O* -11916.042 -0.362
OH* -11899.149 0.830
O* -11882.980 1.298
H* -11464.540 -0.619
COOH* -12490.256 -1.487
Co-Adsorbed Species (1NN) DFT Energy (eV) Form. Energy (eV)
CO*+CO* -12629.394 -3.595
OH*+H* -11915.448 0.233
O*+H* -11899.102 0.877
CO*+OH* -12489.950 -1.181
CO*+O* -12473.424 -0.357
Transition States DFT Energy (eV) Form. Energy (eV)
TS1: H2O*→OH*+H* -11915.265 0.416
TS2: OH*→O*+H* -11898.209 1.770
TS3: CO*+OH*→COOH* -12489.544 -0.776
TS4: COOH*→CO2+H* -12489.404 -0.635
TS5: CO*+O*→CO2 -12472.435 0.632

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